Core Shield Generators are linked into five networks: A, B, C, D, & E.
Networks B, C, D, & E will destabilize and self destruct once any shield generator in those networks is destroyed.
Network A is stronger however and all generators (except the last) must be destroyed by human forces. Once there is only a single Network A Shield Generator left it will self-destruct as there are no other shield generators left for it to network with.
The status of all five networks can be seen in the Stats -> Objectives window.
Core Shield Generators are either enabled or disabled for both AIs at the same time and can be found on the Ships page rather then the AI Plots page in the Game Lobby.
While having to take systems to destroy these Core Shield Generators does increase the number of systems you have to take before being able to attack the AI Homeworlds, the Core Shield Generators are usually located on high value worlds, especially networks A and B, and so you will often find that taking these systems would be worth it for other reasons.

These structures are unarmed so are easy to destroy once you are able to. The only trick they have is that because they are linked into a network, they are invulnerable until the planet they are on is captured by human forces, weakening them so they can be damaged.